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A120934
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Least prime p such that the interval [p,p+log(p)] contains n primes.
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2
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2, 11, 457, 3251, 165701, 10526557, 495233351, 196039655873, 10687033762033, 79006533276941, 4313367040646743, 1740318019946551931
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OFFSET
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1,1
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COMMENTS
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Soundararajan states that, on average, there is one prime in the interval [k,k+log(k)] for any number k. Is there an upper limit to the number of primes in such an interval? Not if the prime k-tuple conjecture is true, in which case a(n) exists for all n. Note that a(n) > e^A008407(n). See A120935 for the largest prime in the interval.
a(n) begins a sequence of n primes whose prime pattern is one of the patterns in the n-th row of A186634. For example, the sequence of four consecutive primes beginning with 3251 is (3251, 3253, 3257, 3259), which has pattern (0, 2, 6, 8), which is in the 4-th row of A186634.
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LINKS
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Table of n, a(n) for n=1..12.
K. Soundararajan, The distribution of prime numbers
Eric Weisstein's World of Mathematics, Prime k-Tuple Conjecture
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FORMULA
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This sequence grows superexponentially; a weak lower bound is a(n) >> (log n)^n. It seems that a(n) > n^n. - Charles R Greathouse IV, Apr 18 2012
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EXAMPLE
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a(2)=11 because p=11 is the first prime with log(p)>2 and 11+2 is prime.
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MATHEMATICA
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i=1; Table[While[p=Prime[i]; PrimePi[p+Log[p]]-PrimePi[p]+1< n, i++ ]; p, {n, 5}]
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CROSSREFS
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Cf. A120936 (number of primes in the interval [n, n+log(n)]), A020497.
Sequence in context: A012950 A012979 A013109 * A000886 A128855 A206846
Adjacent sequences: A120931 A120932 A120933 * A120935 A120936 A120937
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KEYWORD
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hard,nonn,nice
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AUTHOR
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T. D. Noe, Jul 21 2006
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EXTENSIONS
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a(12) from Donovan Johnson, Apr 18 2012
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STATUS
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approved
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